Multifractal analysis of implied volatility in index options

In this paper, we analyze the statistical and the non-linear properties of the log-variations in implied volatility for the CAC40, DAX and S&P500 daily index options. The price of an index option is generally represented by its implied volatility surface, including its smile and skew properties. We utilize a Lévy process model as the underlying asset to deepen our understanding of the intrinsic property of the implied volatility in the index options and estimate the implied volatility surface. We find that the options pricing models with the exponential Lévy model can reproduce the smile or sneer features of the implied volatility that are observed in real options markets. We study the variation in the implied volatility for at-the-money index call and put options, and we find that the distribution function follows a power-law distribution with an exponent of 3.5 ≤ γ ≤ 4.5. Especially, the variation in the implied volatility exhibits multifractal spectral characteristics, and the global financial crisis has influenced the complexity of the option markets.